Atomic force microscope apparatus

ABSTRACT

An object of the present invention is to provide an atomic force microscope apparatus allowing tracking errors to be made as close to zero as possible to reduce images obtained through high-speed scanning from being degraded. To accomplish the object of the present invention, the present invention provides an atomic force microscope apparatus imaging a surface topography of a sample in a contact mode, the apparatus including a cantilever having a probe interacting with the sample surface via an atomic force and being subjected to a deflection by the atomic force, laser light provision means for allowing first laser light to enter the cantilever, light detection means, a controller estimating the surface topography of the sample surface, and data storage means for recording the estimated surface topography.

TECHNICAL FIELD

The present invention relates to an atomic force microscope apparatus.

BACKGROUND ART

An atomic force microscope (AFM) is an apparatus using a probe toperform scanning along the surface of a sample to measure displacementof a cantilever caused by recesses and protrusions on the surface andforming the measured displacement into an image of the surface, thusmeasuring the surface of the sample on a nano scale. In general, a forceis inevitably exerted between two objects (in this case, a probe tip anda sample) arranged in proximity to each other. Thus, since the AFMmeasures a variation in force caused by recesses and protrusions on thesurface of the sample, as the displacement of the cantilever, the AFM inprinciple imposes no restrictions on the sample. Consequently, the AFMcan observe even the structure of an insulator surface which an STM(scanning tunnel microscope) cannot observe.

The accuracy of observation images obtained with the AFM depends on theperformance of a feedback controller. With classical control such as PIcontrol that is a conventional control scheme, the relevant frequencyband is limited by the resonance frequency of the mechanism. Thus,various efforts have been made to improve the performance of thefeedback controller.

For example, the following have been introduced into the control of theAFM in order to allow the z piezo elements in the AFM to be quicklydriven: a counter balance method and an active damping method(Non-Patent Document 17 and Non-Patent Document 18), a method offeedforward-compensating information for every shift mode or line(Non-Patent Document 13), and Q value control for a cantilever. However,most of these methods are based on the classical control such as the PIcontrol and implemented in analog circuits (Non-Patent Document 20).Alternatively, an H_(∞) loop shaping method (Non-Patent Document 21), anadaptive control method (Non-Patent Document 15), and the like may beapplied. However, the feedback control system may inevitably berestricted by a Bode's integral theorem.

AFM operation schemes include a contact mode, a non-contact mode, and atapping mode. The contact mode is based on a contact scheme in which aprobe is contacted with the sample surface for scanning. The non-contactmode is based on a non-contact scheme in which the probe is notcontacted with the sample surface and the surface topography is measuredbased on a variation in the oscillation frequency of a cantilever. Thetapping mode is based on a periodic contact scheme in which the probe isperiodically contacted with the sample surface to measure the surfacetopography based on a variation in the oscillation amplitude of thecantilever (see Non-Patent Documents 2 and 3). An analysis method forthe surface topography based on the contact mode generally controls apiezo Z axis so as to maintain the displacement of the cantileverconstant and records a manipulating quantity u(t) for the axis as asurface topography. However, in the analysis method, the relevantfrequency band may disadvantageously be limited by the resonantfrequency of the mechanism as described above.

Thus, Non-Patent Document 1 proposes a method for estimating the surfaceof a sample using a disturbance observer (the method is simply referredto as STO). In connection with this control method, Non-Patent Document1 clarifies that modeling a plant allows the surface topography of anobject to be estimated using an estimation mechanism similar to thedisturbance observer. Thus, for the STO, the relevant frequency band isnot limited by a closed loop. Consequently, the STO is demonstrated tobe more advantageous than the conventional method even though themanipulating quantity u(t) does not actually track the surfacetopography.

In Non-Patent Document 10, hardware is improved to increase theoperation speed of the AFM. However, in embodiments according to thepresent invention described below, control is improved to preventpossible degradation of images obtained with the AFM through high-speedscanning.

-   [Non-Patent Document 1] “Study of Production and Control of    Nano-scale Servo Apparatus for Atomic Force Microscope”, Industrial    Instrumentation and Control Workshop of the Institute of Electrical    Engineers of Japan, IIC-06-132, p. 1-6 (2006)-   [Non-Patent Document 2] “Introduction to Nano-Probe Technique”,    Kogyo Chosakai Publishing, Inc. (2001)-   [Non-Patent Document 3] “Scanning Probe Microscope”, MARUZEN Co.,    Ltd.-   [Non-Patent Document 4] “Introduction to System Control Theories”,    Jikkyo Shuppan Co., Ltd.-   [Non-Patent Document 5] “Perfect Tracking Control Method Using    Multirate Feedforward Control”, Collection of Papers for the Society    of Instrumentation and Control Engineers, 36, p. 766-772 (2000)-   [Non-Patent Document 6] “PRO Compensation of Magnetic Disk Apparatus    Based on Switching Control and PTC, Industrial Instrumentation and    Control Workshop of the Institute of Electrical Engineers of Japan,    IIC-04-69, p. 13-18 (2004)-   [Non-Patent Document 7] “Harmonic analysis based modeling of tapping    mode AFM”, Processings of the American Control Conference, p.    232-236 (1999)-   [Non-Patent Document 8] “System Identification for Control Based on    MATLAB”, Tokyo Denki University Press (1996)-   [Non-Patent Document 9] “Advanced System Identification for Control    Based on MATLAB”, Tokyo Denki University Press (2004)-   [Non-Patent Document 10] “High-Speed Video Rate AFM”,    Instrumentation and Control, Vol. 45, No. 2, p. 99-104 (2006)-   [Non-Patent Document 11] “Proposal of Nano-Scale Servo for Atomic    Force Microscope Based on Surface Topography Learning with PTC”,    Industrial Instrumentation and Control Workshop of the Institute of    Electrical Engineers of Japan, IIC-07-52, p. 7-12 (2007)-   [Non-Patent Document 12] “Study of Production and Control of    Nano-scale Servo Apparatus for Atomic Force Microscope”, Industrial    Instrumentation and Control Workshop of the Institute of Electrical    Engineers of Japan, IIC-06-132, p. 1-6 (2006)-   [Non-Patent Document 13] “Robust Two-Degree-of-Freedom Control of an    Atomic Force Microscope”, Asian Journal of Control, Vol. 6, Bo.    2, p. 156-163 (2004)-   [Non-Patent Document 14] “Robust Control Approach to Atomic Force    Microscopy”, Conf. Decision Contr., p. 3443-3444 (2003)-   [Non-Patent Document 15] “On Automating Atomic Force Microcscopes:    An Adaptive Control Approach”, Conf. Decision Contr., p. 1574-1579    (2004)-   [Non-Patent Document 16] “Digital control of repetitive errors in    disk drive system”, IEEE Contr. Syst. Mag., Vol. 10, No. 1, pp.    16-20 (1990)-   [Non-Patent Document 17] Rev. Sci. Instrum., 76, 053708 (2005)-   [Non-Patent Document 18] Proc. Natl. USA. Sci. USA, 98, 12468 (2001)-   [Non-Patent Document 19] Phys. Rev. Lett. 90, 046808 (2003)-   [Non-Patent Document 20] “Proposal of Surface Topography Observer    for Tapping Mode AFM”, IIC-07-119 (2007)-   [Non-Patent Document 21] “Robust Control Approach to Atomic Force    Microscopy”, Conf. Decision Contr., p. 3443-3444 (2003)-   [Non-Patent Document 22] “Proposal of Surface Topography Learning    Observer for Contact Mode AFM”, IIC-07-117, p. 7-12 (2007)-   [Non-Patent Document 23] “Zero Phase Error Tracking Algorithm for    Digital Control”, Trans. ASME, Journal of Dynamic Systems,    Measurement, and Control, Vol. 109, p. 65-68 (1987)-   [Non-Patent Document 24] “Zeros of sampled system”, Automatica, 20,    1, p. 31-38 (1984)-   [Non-Patent Document 25] “Perfect Tracking Control Method Based on    Multirate Feedforward Control”, Trans. SICE, Vol. 36, No. 9, p.    766-772 (2000)

DISCLOSURE OF THE INVENTION

However, the STO disadvantageously precludes accurate measurements whenthe plant suffers large modeling errors.

To solve the above-described problems, the present invention provides anatomic force microscope apparatus imaging a surface topography of asample in a contact mode, the apparatus comprising a cantilever having aprobe interacting with the sample surface via an atomic force and beingsubjected to a deflection by the atomic force, laser light provisionmeans for allowing first laser light to enter the cantilever, lightdetection means for detecting second laser light corresponding to thefirst laser light reflected by the cantilever, a piezo element on whichthe sample is placed, a controller inputting an input voltage to thepiezo element to control the distance between the sample surface and theprobe, detecting the deflection of the cantilever as an output voltagebased on a relative change in intensity of the second laser light, thenduring a forward scan, measuring and storing the surface topography, andduring a backward scan of the same line as that for the forward scan,using the stored surface topography for control to estimate the surfacetopography of the sample surface based on the output voltage, and datastorage means for recording the estimated surface topography.

The present invention can make tracking errors as close to zero aspossible to reduce images obtained through high-speed scanning frombeing degraded.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a diagram schematically showing an atomic force microscope(AFM) according to the present invention;

FIG. 2 is a diagram showing an interactive force exerted between the tipof a cantilever and a sample surface based on a contact mode;

FIG. 3 is a block diagram showing the flow of a signal in the AFMaccording to the present invention;

FIG. 4 is a diagram showing multirate control;

FIG. 5 is a diagram showing the procedure of control;

FIG. 6 is a diagram showing a surface scan path;

FIG. 7 is a diagram showing a surface scan path;

FIG. 8 is a Bode diagram;

FIG. 9 is a diagram showing a temporal variation in output voltage;

FIG. 10 is a diagram showing a loop transfer function;

FIG. 11 is a block diagram of a surface topography observer (STO);

FIG. 12 is a block diagram of surface topography learning with PTC(STL-PTC);

FIG. 13 is a diagram showing a signal generator for error signals;

FIG. 14 is a diagram showing a complementary sensitivity function;

FIG. 15 is a diagram showing the results of simulation of an outputvoltage obtained when the AFM according to the present invention adoptsthe conventional method to scan a rectangular wave-like sample surface;

FIG. 16 is a diagram showing a frequency response from Q(s) in the STO;

FIG. 17 is a diagram showing the results of simulation of an outputvoltage obtained when the AFM according to the present invention adoptsthe STO to scan the rectangular wave-like sample surface;

FIG. 18A is a diagram showing the shape of a grating element;

FIG. 18B is a diagram showing the shape of the grating element;

FIG. 19 is a diagram showing superimposed error signals in one image ofa sample measured with the AFM according to the present invention;

FIG. 20 is a diagram showing superimposed error signals in one image ofthe sample measured with the AFM according to the present invention;

FIG. 21 is a diagram showing a standard deviation in error signal;

FIG. 22 is a diagram showing superimposed error signals in one image ofa sample measured with the AFM according to the present invention;

FIG. 23 is a diagram showing superimposed error signals in one image ofthe sample measured with the AFM according to the present invention;

FIG. 24 is a diagram showing a standard deviation in error signal;

FIG. 25 is a diagram showing an image of a sample measured with the AFMaccording to the present invention;

FIG. 26 is a diagram showing an image of the sample measured with theAFM according to the present invention;

FIG. 27 is a diagram showing an image of the sample measured with theAFM according to the present invention;

FIG. 28 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 29 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 30 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 31 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 32 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 33 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 34 is a diagram showing an image of the sample measured with theAFM according to the present invention;

FIG. 35 is a diagram showing an image of the sample measured with theAFM according to the present invention;

FIG. 36 is a diagram showing an image of the sample measured with theAFM according to the present invention;

FIG. 37 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 38 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 39 is a diagram showing the frequency of height of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 40 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 41 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 42 is a diagram showing the sectional waveform of recesses andprotrusions on the surface of the sample measured with the AFM accordingto the present invention;

FIG. 43 is a Bode diagram;

FIG. 44 is a Bode diagram;

FIG. 45 is a diagram showing a loop transfer function;

FIG. 46 is a diagram showing a loop transfer function;

FIG. 47 is a block diagram of improved surface topography learning withPTC (STL-PTC);

FIG. 48 is a diagram showing a signal generator;

FIG. 49 is a block diagram of a surface topography learning observer(STLO);

FIG. 50 is a diagram showing a frequency response from a Q filter;

FIG. 51 is a diagram showing a frequency response from the Q filter;

FIG. 52 is a diagram showing the results of simulation of scanning of arectangular wave-like sample surface;

FIG. 53 is a diagram showing the results of simulation of scanning ofthe rectangular wave-like sample surface;

FIG. 54 is a diagram showing the results of simulation of scanning ofthe rectangular wave-like sample surface;

FIG. 55 is a diagram showing the results of simulation of scanning ofthe rectangular wave-like sample surface;

FIG. 56 is a diagram showing the shape of a grating element;

FIG. 57 is a diagram showing standard deviations;

FIG. 58 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 59 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 60 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 61 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 62 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 63 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 64 is a diagram showing an image of a sample measured with the AFM;

FIG. 65 is a diagram showing an image of the sample measured with theAFM;

FIG. 66 is a diagram showing an image of the sample measured with theAFM;

FIG. 67 is a diagram showing an image of the sample measured with theAFM;

FIG. 68 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 69 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 70 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 71 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 72 is a diagram showing an image of a sample measured with the AFM;

FIG. 73 is a diagram showing an image of the sample measured with theAFM;

FIG. 74 is a diagram showing an image of the sample measured with theAFM;

FIG. 75 is a diagram showing an image of the sample measured with theAFM;

FIG. 76 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 77 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 78 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 79 is a diagram showing the sectional waveform of the samplemeasured with the AFM;

FIG. 80 is a diagram showing frequency characteristics;

FIG. 81 is a diagram showing frequency characteristics;

FIG. 82 is a diagram showing frequency characteristics;

FIG. 83 is a diagram showing frequency characteristics;

FIG. 84 is a diagram showing a control mechanism according to thepresent embodiment;

FIG. 85 is a diagram showing the results of simulation;

FIG. 86 is a diagram showing the results of simulation;

FIG. 87 is a diagram showing the results of simulation;

FIG. 88 is a diagram showing the results of simulation;

FIG. 89 is a diagram showing frequency characteristics;

FIG. 90 is a diagram showing frequency characteristics;

FIG. 91 is a diagram showing frequency characteristics;

FIG. 92 is a diagram showing frequency characteristics;

FIG. 93 is a diagram showing an image of a sample measured with the AFM;

FIG. 94 is a diagram showing an image of the sample measured with theAFM;

FIG. 95 is a diagram showing an image of the sample measured with theAFM;

FIG. 96 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 97 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 98 is a diagram showing a waveform in which error signals aresuperimposed on one another;

FIG. 99 is a diagram showing the results of simulation;

FIG. 100 is a diagram showing the results of simulation;

FIG. 101 is a diagram showing the results of simulation;

FIG. 102 is a diagram showing the results of simulation;

FIG. 103 is a diagram showing a signal generator;

FIG. 104 is a diagram showing the results of simulation;

FIG. 105 is a diagram showing the results of simulation;

FIG. 106 is a diagram showing the results of simulation;

FIG. 107 is a diagram showing the results of simulation;

FIG. 108 is a diagram showing the results of simulation; and

FIG. 109 is a diagram showing the results of simulation.

BEST MODE FOR CARRYING OUT THE INVENTION First Embodiment

FIG. 1 is a schematic diagram showing an AFM 100 according to thepresent invention by way of example. In FIG. 1, in the AFM 100, a probe102 attached to a cantilever 101 performs scanning along a samplesurface 103 to measure deflection of the cantilever 101 caused by anatomic force exerted between the sample surface 103 and the probe 102 aswell as distortion of the cantilever 101 caused by a friction forceexerted between the sample surface 103 and the probe 102. The structureof the sample surface 103 is measured on a nano-scale.

In the AFM 100 shown in FIG. 1, laser light provision means 110 allowslaser light 104 to obliquely enter the rear surface of the cantilever101. Then, a change in the reflection angle of the laser light 104caused by displacement of cantilever 101 resulting from deflection anddistortion thereof is detected based on a relative change in theintensity of laser light 106 entering a four-piece photodiode 105.Finally, the AFM 100 can detect the deflection and distortion of thecantilever 101 based on a change in the intensity of the laser light 106to measure the structure of the sample surface 103.

The aspect shown herein is only an example. The apparatus detecting arelative change in the intensity of the laser light 106 is not limitedto the four-piece photodiode but may be light detection means capable ofdetecting a relative change in the intensity of the laser beam.Alternatively, for example, a visible-light semiconductor laser may beused as the laser light provision means 110.

In the AFM 100 shown in FIG. 1, a controller 108 controls a piezo 107 soas to maintain the displacement of the cantilever 101 constant. Anoutput from the controller 108 is converted, and the converted output isrecorded in the data storage means 109 as the surface topography of thesample surface 103.

(1: Light Detection Scheme for the AFM)

Schemes of measuring the “displacement of the cantilever” caused bysurface recesses and protrusions include a scheme of measuring theinterference of the laser light (light interference scheme) and a lightleverage scheme of measuring a change in the reflection angle of thelaser light caused by the displacement of the cantilever. The presentembodiment uses the light leverage scheme, which is more common.

The light leverage scheme measures a relative change in the intensity oflight entering diodes 1 to 4. The light leverage scheme is based on thedeflection of the tip of the leverage (the tip of the cantilever) in a Ydirection and the twist of the tip of the leverage in an X direction.For the deflection, a relative change (1+2)−(3+4) is detected. For thetwist, a relative change (1+4)−(2+3) is detected. In particular, for thetwist, the friction force is measured. In this case, the microscope iscalled an FEM (Friction Force Microscope) instead of the AFM. The AFMaccording to the present embodiment measures only the deflection andthus corresponds to the former AFM detection method.

(2: Modeling in the Contact Mode)

If a disturbance observer is used for the AFM or in order to design thecontroller, a controlled object needs to be modeled. Thus, a model ofthe AFM is based on the cantilever and the interaction between the probeand the sample. Such a model as shown in FIG. 2 is adopted (seeNon-Patent Documents 1 and 7). As a measurement mode, the contact modeis used for measurement in order to simplify the modeling.

The moment when a spring comes into contact with the sample, the springhas a natural length. Thus, the spring coefficient of a spring 203 isdefined as k₁, and the natural length of the spring 203 is defined asL₁₀. The spring coefficient of a spring 201 is defined as k₂, and thenatural length of the spring 201 is defined as L₂₀. Furthermore,reference character (b) denotes the damper coefficient of a frictionforce generation source 205. The current lengths of the springs 203 and201 are defined as L₁ and L₂, respectively. Reference character (u)denotes a manipulating quantity for the piezo. Reference character (d)denotes the recesses and protrusions of the sample. In this case, theinteraction between the sample and the cantilever is expressed as shownin FIG. 2. F(t) denotes a force that the cantilever receives from thespring 201, that is, an atomic force from the sample.

Based on the model shown in FIG. 2, a motion equation for a cantileverwith a mass (m) is expressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 1} \right\rbrack & \; \\{y = \frac{k_{2} \times {g\left( {u + d} \right)}}{{m\; s^{2}} + {bs} + \left( {k_{1} + k_{2}} \right)}} & (1)\end{matrix}$Non-Patent Document 1 indicates that based on the model, the recessesand protrusions of the sample can be modeled as an input disturbance. Inactual experiments, the displacement (y) of the cantilever is measuredusing the photodiode and laser light. Thus, the displacement (y) isdetermined by multiplying a transfer function for a plant by a givengain (g). The relationship between the gain (g) and the output will bedescribed in the next chapter. The details of a method for derivingFormula (1) are described in, for example, Non-Patent Document 1.(3: System Identification)(3-1: Construction of the AFM)

The AFM according to the present invention may be constructed byconnecting an interface for required input signals to a JSPM-5200manufactured by JEOL Ltd. and using a controller board such as a Dspace1104 to improve an algorithm and hardware for a control system. Thedetails of the algorithm for the control system are described in, forexample, Non-Patent Documents 8 and 9.

FIG. 3 is a block diagram showing the flow of signals in the AFMaccording to the present invention. In FIG. 3, when the sample isscanned, the displacement of the cantilever 101 is output by the PD(PhotoDiode) 105. This signal is converted by an AD 305, and theresulting signal is input to a DSP (Digital Signal Processor) as y[i].Furthermore, in the DSP, a DA 301 converts a manipulating quantity u[i],and an amplifier 302 amplifies the converted manipulating quantity u[i].The amplified signal is applied to the PZT (piezo) 107 as a drivingvoltage. A gain obtained by the amplifier is K_(g)=20. A driving voltagefor the PZT is K_(PZT)=15.59 [nm/V]. Furthermore, the gain of AD/DA inthe DSP is adjusted to 1.

An output x [V] from the PD is output as a voltage indicating thedisplacement of the cantilever and varying depending on a force curve.Thus, to convert the displacement x [V] of the cantilever into y [nm], arelational expression for the driving voltage for the PZT and the outputfrom the PD is determined based on the force curve. As a result,K_(PD)=3.61×10⁻² [V/nm].

The K_(PD) is determined from the first-order approximation ofmeasurement data of the force curve obtained by the JSPM-5200. The forcecurve is described in, for example, Non-Patent Document 1.

(3-2 Identification Experiments)

In system identification experiments, a model is estimated using a leastsquares method based on experimentally obtained I/O data. Foridentification conditions, an M-sequence signal is used as anidentification input (pseudo disturbance). An ARX model is used toestimate a model (see Non-Patent Document 8). With the least squaresmethod, transfer functions at discrete times are each expressed by asecond-order denominator and a first-order numerator. A zeroth-orderhold is used to convert the transfer functions into a continuous time. Atransfer function from formula (1) can be expressed as formula (2). Itshould be noted that the M-sequence signal cannot be provided directlyto the cantilever, so that in FIG. 3, the M-sequence signal is input tothe PZT via the DA 301, with estimation performed based on an outputfrom the AD 305.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 2} \right\rbrack & \; \\{{P(s)} = \frac{10.034 \times 10^{9}}{s^{2} + {8219s} + {1.274 \times 10^{9}}}} & (2)\end{matrix}$

FIG. 8 is a Bode diagram in which the frequency characteristics of aplant in the above-described model based on formula (2) are comparedwith those identified by a servo analyzer. The identified plant exhibitsa significant resonance at 5,610 [Hz] and has a high gain even in a lowfrequency region.

Furthermore, FIG. 9 shows a comparison of a temporal variation involtage observed with the AFM according to the present invention, with atemporal variation in voltage provided by the above-described model.FIG. 9 shows that model outputs allow measurement outputs (actualoutputs) to be reproduced to some degree.

(4: Design of the Controller)

(4-1: Design of the Controller Using the Conventional Method)

A controller used for comparison with the proposed method is provided inan actual product. The controller is defined as a conventional method.An expression for the controller is given as follows.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 3} \right\rbrack & \; \\{{C(s)} = {\frac{\omega_{c}}{s + \omega_{c}}k_{p}}} & (3)\end{matrix}$

Tuning was performed until the experimental apparatus started tooscillate as described in the manual. The controller was thus designedwith a proportional gain k_(p) set to 64 and (O set to 2πf_(c)(f_(c)=0.5 [Hz]). FIG. 10 shows a loop transfer function for the plantand the controller. FIG. 10 shows that the conventional method has acutoff frequency of 252 [Hz], indicating that proportional control and alow pass filter can deal with up to this frequency band. Furthermore, again margin is 14.3 [dB], and a phase margin is 89.5 [deg].

(4-2: Surface Topography Observer (STO))

Based on the manipulating quantity (u) and the measurement output (y),recesses and protrusions (d) on the sample surface, corresponding to adisturbance, are estimated using an observer. The estimated value:{circumflex over (d)}  [Expression 4]is obtained by passing a signal passed through the inverse model of anominal plant:P _(n) ⁻¹(s)  [Expression 5]and from which the manipulating quantity u(t) is subtracted, through alow pass filter Q(s) for the cutoff frequency ω_(c). FIG. 11 is a blockdiagram of estimation in the STO. In FIG. 11, an estimation block 1102in an estimation block 1100 is characterized by being implemented as adisturbance observer for an open loop independently of a feedback loop1101. In FIG. 11, a surface topography 703 is considered to be an inputend disturbance d(t) at time (t). The estimated value for the surfacetopography is obtained from an output 1104.

In Non-Patent Document 1, this special disturbance observer is referredto as a surface topography observer (STO). The STO is composed of anopen loop and is thus not limited to the frequency band of a closedloop. The frequency band of Q(s) can be increased up to a Nyquistfrequency. Thus, even if u(t) does not actually track recesses andprotrusions, that is, even if a tracking error (e) is not zero, providedthat the nominal plant has no modeling error, the recesses andprotrusions (d) of the sample can be accurately estimated at a frequencyequal to or lower than ω_(c). Furthermore, a frequency response from theobserver is:[Expression 6]{circumflex over (d)}/d=P(s)×P _(n) ⁻¹(s)×Q(s)≈Q(s)  (4)Based on a second-order low pass filter, Q(s) can be expressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 7} \right\rbrack & \; \\{{Q(s)} = \left( \frac{\omega_{c}}{s + \omega_{c}} \right)^{2}} & (5)\end{matrix}$(4-3: Design of the Controller According to the Proposed Method)

The STO described in the preceding chapter is composed of an open loop.Thus, increasing the frequency band of Q(s) above that of a closed loopis considered to be preferable. However, since the observer is an openloop, when the tracking capability of u(t) is significantly degraded, anerror in the modeling of the plant increases owing to a Lennard-Jonespotential (see Non-Patent Document 1). This significantly degradesrobust stability. Thus, adjusting the tracking error (e) to 0 resultingin a modeling error is expected to avoid the disadvantages of the STO.Consequently, the present embodiment applies a PTC method to allow atarget trajectory (described below) generated from the learned trackingerror (e) to be perfectly tracked. Therefore, the tracking error (e) isreduced in a feedforward manner, thus improving the control performance.

As shown in FIG. 4, the PTC method corresponds to atwo-degree-of-freedom control system in which a sampling period T_(r)for the target trajectory is different from a control period T_(u) forthe target trajectory. In the control method, during one sampling periodT_(r) for a command value, a control input is switched (n) times at theintervals of T_(u). In this case, (n) denotes the order of the plant.Normally, when the reverse system of the plant is constructed at asingle rate, the feedforward controller becomes unstable under theeffect of unstable zeroes generated when the plant based on a linearcontinuous-time system is discretized at a short sampling period.Therefore, multirate control allows the feedforward controller to createthe stable reverse system of the plant (see Non-Patent Document 5).

(4-3-1: Discretization of the Controlled Object)

It is assumed that a modeled second-order controlled object isdiscretized. When a state variable is defined as (x), a state equationfor the continuous-time system is given as follows:[Expression 8]{dot over (x)}=A _(c) x(t)+b _(c) u(t)  (6)[Expression 9]y(t)=c _(c) x(t)  (7)

A state equation that discretizes the controlled object at the shortersampling period T_(u) is expressed as:[Expression 10]x[k+1]=A _(s) x[k]+b _(x) u[k]  (8)[Expression 11]y[k]=c _(x) x[k]  (9)

In this case, x[k]=x(kT_(u)), and[Expression 12]A _(c) =e ^(A) ^(c) ^(T) ^(u) , b _(c)=^(T) ^(u) e ^(A) ^(c) ^(τ) b _(c)dτ  (10)(4-3-2: Surface Topography Learning with PTC)

Now, description will be given of surface topography learning with PTCfor an AFM in which a cantilever is reciprocated over a sample tomeasure the surface topography of the sample; the surface topographylearning with PTC involves measuring and storing a tracking error duringa forward scan and using the stored tracking error to increase trackingaccuracy during a backward scan.

First, when the normal PTC method is used to control the AFM, a setpoint corresponds to the balanced position of the cantilever and is thuszero. Thus, scanning in the current condition results in only feedbackcontrol, corresponding to the conventional control method. According toNon-Patent Document 6, although a repetitive PTC method of reducing apossible periodic disturbance for every sample point is applicable,selecting only samples with periodic surface topographies formeasurements by the AFM is impossible.

Thus, focusing on the scan method for the AFM, the surface topographylearning with PTC (STL-PTC) learns and controls the surface of thesample during the forward and backward scans. In the present embodiment,a grating element is used to observe the sample. However, it should benoted that any element other than the grating element are applicable.

FIG. 12 is a block diagram of STL-PTC. In the estimation blocks of theSTL-PTC shown in FIG. 12, a surface topography 1201 is input, and anestimated surface topography is stored in data storage means 1202.Furthermore, reference numeral 1203 denotes a signal generator, andreference numeral 1204 denotes a switch. The manipulating quantity (u)indicates surface topography data. Since the set point is zero, anoutput (voltage) (y) is an error signal (e). A smaller value of theerror signal (e) means a more accurate image resulting from the currentmanipulating quantity (u). That is, the error signal (e) corresponds toa tracking error used for the surface topography learning with PTC.

FIG. 6 shows a surface scan path of the probe of the AFM according tothe present invention. As shown in FIG. 6, to measure a sample in animage, a CPU mounted in the AFM according to the present invention readsand executes a program stored in a storage device to perform theSTL-PTC. The probe performs a rightward X-direction scan over a scanwidth from a start position. The probe then performs a leftwardX-direction scan along the scan path to return to the scan startposition. Then, the probe similarly performs a scan in the Y direction.Thus, a surface scan is achieved.

The rightward scan in the X direction is called a forward scan (FWS).The leftward scan in the X direction is called a backward scan (BWS).The two scans allow the image to be measured. Provided that the probefollows the same path both during the FWS and during the BWS, errorsignals appearing during the scans are ideally exactly the same.

Thus, in the surface topography learning with PTC, during the FWS, errorsignals are measured and stored and then learned, and based on thelearned error signals, learning control is performed so as to cancel apossible error signal during the BWS. Consequently, error signals (e)(tracking errors) for feedback control can be reduced to improve thetracking capability.

FIG. 5 shows a procedure of control for the AFM according to the presentinvention. As shown in FIG. 5, an X scan waveform is triangular. Eachimage is measured during an FWS 501 and during a BWS 502. During theFWS, a switch 1 (SW1) in FIG. 13 is kept on for T (=the number N_(m) ofstages in the memory×the sampling period T_(y) of an output signal)seconds (switch-on and switch-off are shown by the height of a graphshown at reference numeral 503). Error signals are stored in a signalgenerator composed of a stack memory 1301 shown in FIG. 13. During theBWS, a switch 2 (SW2) is turned on to allow the signal generator togenerate a target trajectory allowing the error to be adjusted to 0.Then, the PTC adjusts the error to 0. Thus, the N_(m) memory rows canserve as a feedforward compensator. This enables possible error signalsto be reduced for every sample point. However, during the FWS, thesignal generator provides no output.

In FIG. 5, reference numerals 504, 506, and 508 denote learningprocesses. Reference numerals 505, 507, and 509 denote controlprocesses.

Here, since the controlled object is of a second order, when a statevariable (x) is:x=[y,{dot over (y)}]  [Expression 13]the signal generator can be designed for error signals as shown in FIG.13 (see Non-Patent Document 6). However, r[i] and{dot over (r)}[i]  [Expression 14]are as follows:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 15} \right\rbrack & \; \\{{r\lbrack i\rbrack} = {{- \frac{z}{z^{N_{m}} - 1}}{y\lbrack i\rbrack}}} & (11) \\\left\lbrack {{Expression}\mspace{14mu} 16} \right\rbrack & \; \\{{\overset{.}{r}\lbrack i\rbrack} = \frac{{r\left\lbrack {i + 1} \right\rbrack} - {r\left\lbrack {i - 1} \right\rbrack}}{2T_{y}}} & (12)\end{matrix}$

Furthermore, the STO is an open loop, and a modeling error may thusdegrade the robustness of the STO. However, the proposed methodcompensates for the degraded robustness by the feedback control.

(4-3-3: PTC (Perfect Tracking Control) Method)

The PTC method is based on multirate control provided by a feedforwardcontroller and a feedback controller to achieve perfect tracking incontrast to a method based on single-rate control (see Non-PatentDocument 5).

Now, the design of the feedforward controller will be described. Sincethe order (n) of the plant is 2, formulas (8) and (9) are used for thesecond sample. Furthermore, when time (t)=iT_(r)=kT_(u), the followingare given.[Expression 17]x[i+1]=Ax[i]+Bu[i]  (13)A=A _(s) ²  [Expression 18]B=[A _(s) b _(s) ,b _(s)]  [Expression 19]

Based on a discrete-time state equation for the controlled objectexpressed by formula (13), a stable reverse system is obtained which isexpressed by:[Expression 20]u[i]=B ⁻¹(I−z ⁻¹ A)x[i+1]  (14)

Provided that the controlled object in formula (8) is controllable, theregularity of a matrix B is ensured. Furthermore, in formula (14), polesare present at the origin of a (z) plane, indicating that the reversesystem is stable. Thus, a predicted value r [i]=x_(d)[i+1] for thetarget trajectory of the controlled object is given as a reference valuer[i]. Then, a feedforward control output is given as shown in formula(15). As a result, the nominal plant is perfectly tracked on the samplepoints. Any variation in disturbance or plant is compensated for by afeedback controller C[z]. Furthermore, a nominal output obtained whenthe PTC is established is expressed by formula (16).[Expression 21]u ₀ [i]=B ⁻¹(I−z ⁻¹ A)x _(d) [i+1]  (15)[Expression 22]y ₀ [i]=z ⁻¹ Cx _(d) [i+1]+Du ₀ [i]  (16)(5: Simulation and Experiments)(5-1: Simulation of the STO)

FIG. 14 to FIG. 17 show disturbances estimated by the conventionalmethod and the observer, as simulation of a rectangular wave-likesample.

FIG. 14 shows a complementary sensitivity function obtained by theconventional method.

FIG. 15 is a diagram showing the results of simulation of an outputvoltage obtained when the AFM according to the present invention adoptsthe conventional method to scan a rectangular wave-like sample surface.

FIG. 16 is a diagram showing a frequency response from Q(s) in the STO.

FIG. 17 is a diagram showing the results of simulation of an outputvoltage obtained when the AFM according to the present invention adoptsthe STO to scan the rectangular wave-like sample surface.

FIG. 14 shows that the conventional method limits the poles of theclosed loop to the resonant frequency of the plant. However, FIG. 16shows that the observer of the STO does not depend on the resonantfrequency of the plant but depends on the poles of the low pass filterunless the observer is limited by the Nyquist frequency. Thus, comparedto the manipulating quantity (u) in FIG. 15, the estimated value:{circumflex over (d)}  [Expression 23]in FIG. 17 allows the surface topography (d) of the sample to beaccurately reproduced.

Description will be given below of verification performed throughexperiments using the AFM according to the present invention, to checkthe behavior of the AFM.

(5-2: Observation of the Grating Element)

FIGS. 18A and 18B show the shape and size of a grating element 1801observed with the AFM according to the present invention. Such a gratingelement may be, by way of example, a planar brazed holographic gratingstandard article manufactured by Shimadzu Corporation. The gratingelement shown in FIG. 18A and FIG. 18B is characterized by being shapedlike saw teeth-like grooves. Grating grooves are formed on a glasssubstrate of resin. The grooves are coated with a reflection film of Alor the like.

(5-3: Experiments)

The sampling frequency of a DSP in the AFM according to the presentinvention is set to 10 [kHz] by way of example. Then, the resultsdescribed below were obtained.

FIG. 25 shows an image of the surface of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the conventionalmethod.

FIG. 26 shows an image of the surface the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STO.

FIG. 27 shows an image of the surface the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STL-PTC.

Here, in FIG. 25 to FIG. 27, scanning speed is 32.2 μm/s.

FIG. 28 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the conventional method underthe same conditions as those in FIG. 25.

FIG. 29 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the STO under the sameconditions as those in FIG. 26.

FIG. 30 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the STL-PTC under the sameconditions as those in FIG. 27.

FIG. 31 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the conventional methodunder the same conditions as those in FIG. 25.

FIG. 32 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STO under the sameconditions as those in FIG. 26.

FIG. 33 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STL-PTC under thesame conditions as those in FIG. 27.

FIG. 34 shows an image of the surface of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the conventionalmethod.

FIG. 35 shows an image of the surface of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STO.

FIG. 36 shows an image of the surface of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STL-PTC.

Here, in FIG. 34 to FIG. 36, scanning speed is 161 μm/s.

FIG. 37 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the conventional method underthe same conditions as those in FIG. 34.

FIG. 38 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the STO under the sameconditions as those in FIG. 35.

FIG. 39 is a histogram of the frequency of the height of recesses andprotrusions on the surface of the above-described grating elementobtained by allowing the AFM according to the present invention to scanthe surface of the grating element using the STL-PTC under the sameconditions as those in FIG. 36.

FIG. 40 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the conventional methodunder the same conditions as those in FIG. 34.

FIG. 41 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STO under the sameconditions as those in FIG. 35.

FIG. 42 shows a sectional waveform of the above-described gratingelement obtained by allowing the AFM according to the present inventionto scan the surface of the grating element using the STL-PTC under thesame conditions as those in FIG. 36.

Here, the histograms in FIGS. 28, 30, 37, and 39 relate to the height ofthe sample determined from the manipulating quantity u(t). Thehistograms shown in FIGS. 29 and 38 relate to the height of the sampledetermined from the estimated value:{circumflex over (d)}  [Expression 24]Moreover, FIG. 31 to FIG. 33 and FIG. 40 to FIG. 42 show a waveformobtained by superimposing the 10 cross sections in the images in FIG. 25to FIG. 27, and FIG. 34 to FIG. 36 on one another at intervals of 0.125μm.

Furthermore, in FIG. 25 to FIG. 42, the scanning speed indicates thespeed of scanning in the (x) direction shown in FIG. 6. Scan range is5.5 μm×5.5 μm both for a scanning speed of 32.2 μm/s and for a scanningspeed of 161 μm/s. Time required for the whole scan is about 3 minutesand about 40 seconds, respectively. FIG. 25 to FIG. 27 are enlargedviews of images obtained at a scanning speed of 32.2 μm/s. FIG. 34 toFIG. 36 are enlarged views of images obtained at a scanning speed of 161μm/s. The range for the enlarged images is 1.6 μm×1.6 μm.

With the conventional method, images obtained at an increased scanningspeed as shown in FIG. 34 is significantly degraded compared to thoseobtained at a low scanning speed as shown in FIG. 25. On the other hand,the STO is determined to reduce the degradation of the image compared tothe conventional method when an image obtained at an increased scanningspeed as shown in FIG. 35 is compared with that obtained at a lowscanning speed as shown in FIG. 26.

Furthermore, with an increased scanning speed, the histogram for theconventional method in FIG. 37 shows an extremely small number of heightrates. In contrast, the histogram for the STO shows a relatively largenumber of height rates. Additionally, the sectional waveform for the STOshown in FIG. 41 shows more rugged recesses and protrusions than thatfor the conventional method shown in FIG. 32. However, with theconventional method, the u(t) exhibits a significantly degraded propertyof tracking the surface of the sample. This means that the STO is likelyto increase the modeling error to degrade the image. This is alsoindicated by the fact that images obtained using the STO and shown inFIG. 35 are more significantly degraded than those obtained using theSTL-PTC and shown in FIG. 36 and that the STL-PTC is thus superior tothe STO.

In the histograms shown in FIG. 28 to FIG. 30, the frequency ofprotruding areas increases. This means that the protruding areas areround, so that the amount by which these areas are scanned increases.Furthermore, FIGS. 21 and 24 show a comparison of the conventionalmethod with the STL-PTC, with error signals evaluated in terms of ±3σ.FIGS. 21 and 24 show that “without learning control” indicates theresults for the conventional method, whereas “with learning control”indicates the results for the STL-PTC. FIG. 21 shows the case of ascanning speed of 32.2 μm/s and indicates 4.53% improvement with respectto the height of the recesses and protrusions (60 nm). On the otherhand, FIG. 24 shows the case of a scanning speed of 161 μm/s andindicates 52.5% improvement with respect to the height of the recessesand protrusions (60 nm).

Furthermore, FIGS. 19, 20, 22, and 23 shows superimposed relationshipsbetween error signals and data points (data acquisition points) withinone image.

(6: Summary)

The above-described embodiment of the present invention indicates thedifference between the conventional method and the STO and thus theadvantages of the STO over the conventional method. However,disadvantageously, the STO is not robust to errors in the modeling ofthe plant. Thus, when a point with a rapid change in the recess andprotrusion of the sample is observed, a large modeling error occurs. Atan increased scanning speed, images obtained using the STL-PTC are lessdegraded than those obtained using the STO. This is because the STL-PTCallows possible tracking errors to be reduced by the PTC, while allowingmodeling errors to be compensated for through feedback, enabling thedisadvantages of the STO associated with a rapid change in the recessand protrusion to be avoided.

Second Embodiment

Also in a second embodiment, a model based on a contact mode for theinteraction between a sample surface 103 and the tip 102 of a cantileveris used to give a motion equation for the tip of the cantilever with amass (m), as shown in formula (1).

The method described in Non-Patent Document 2 can be used to convert theabove-described model into one to which the recesses and protrusions onthe sample surface 103 are input. A transfer function for the plantaccording to the present embodiment is identified as follows accordingto a method of system identification described in Non-Patent Document 9.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 25} \right\rbrack & \; \\{{P(s)} = \frac{7.034 \times 10^{9}}{s^{2} + {9219s} + {1.274 \times 10^{9}}}} & (17)\end{matrix}$

FIGS. 43 and 44 show a comparison of the frequency characteristics of aplant based on formula (17) with frequency characteristics identified bya servo analyzer (manufactured by ONO SOKKI Co., Ltd.). The figures showthat the plant identified as shown in formula (17) resonatessignificantly at 5,590 [Hz] and provides a high gain even in a lowfrequency region.

(Internal Configuration of the Experimental Apparatus)

The AFM according to the present embodiment is a special model of aJSPM-5200 manufactured by JEOL Ltd. However, this is only an example,and any AFM is applicable provided that the present embodiment can beincorporated into the AFM. Alternatively, dSPACE1104 may be used tomodify a control mechanism for the AFM so that the control mechanismallows the present embodiment to be implemented.

FIG. 3 is a block diagram showing the flow of signals inside the AFMaccording to the present embodiment.

As shown by reference numeral 310 in FIG. 3, when the sample surface 103is scanned, the displacement of the tip 102 of the cantilever isdetected by a PD (four-piece PhotoDiode) 105 and is output as a signal.The signal is subjected to AD conversion by an AD (AD converter) 305.The resulting signal is input to a DSP (Digital Signal Processor) asy[i].

First, an output x [V] from the PD (four-piece PhotoDiode) 105 isprovided according to a force curve (the relation expression between aforce exerted on the tip of the cantilever and the distance between thecantilever tip and the sample). In the present embodiment, the output x[V] from the PD can be converted into the displacement y [nm] of thecantilever using the conversion expression y [nm]×K_(PD) [V/nm]=x [V].In the present embodiment, based on the force curve, K_(PD)=2.44×10⁻²[V/nm].

In the present embodiment, K_(PD) is determined byfirst-order-approximating the measurement data of the force curve(Non-Patent Document 2) obtained with the JSPM-5200, by way of example.

Furthermore, as shown by reference numeral 320 in FIG. 3, a manipulatingquantity u[i] resulting from DA conversion by the DA (DA converter) 301in the DSP is amplified by an amplifier 302. The amplified manipulatingquantity u[i] is applied to a PZT (piezo) 107 as a driving voltage. Again provided by the amplifier 302 is K_(g)=20, and the rate ofelongation of the PZT subjected to the driving voltage is K_(PZT)=15.59[nm/V]. That is, the calculation u [V]×K_(PZT) [nm/V] allows the u [V]to be converted into the displacement of the piezo.

Additionally, the gain of the AD/DA in the DSP is adjusted to 1.

(Design of the Controller According to the Conventional Method)

The controller according to the conventional method to be compared withthe embodiment of the present invention is a phase delay compensatorused in a product. A transfer function for the controller according tothe conventional method is as shown in formula (3).

Here, the controller according to the conventional method is tuned suchthat the gain margin and the phase margin are 18.8 [dB] and 81 [deg],respectively, and that the proportional gain k_(p)=64 and ω_(c)=2πf_(c)(f_(c)=0.5 [Hz]).

FIGS. 45 and 46 show loop transfer functions for the plant and thecontroller. FIG. 46 shows that the cutoff frequency of the controlleraccording to the conventional method is 177 [Hz].

(Improvement with Improved Surface Topography Learning with PTC)

To avoid the disadvantages of the conventional method and the STO inNon-Patent Document 12, Non-Patent Document 11 proposes the surfacetopography learning with PTC (STL-PTC). The present embodiment uses animproved STL-PTC obtained by improving the surface topography learningwith PTC proposed in Non-Patent Document 11. The learning algorithm ofthe improved STL-PTC is used to perform perfect tracking based on thelearned tracking error (e), thus enabling a surface image observed withthe AFM to be accurately estimated.

Also in the improved STL-PTC, a second-order controlled object modeledas shown in FIG. 2 is discretized using formulas (6) to (16).

FIG. 47 is a block diagram of the improved STL-PTC.

In the estimation blocks of the improved STL-PTC shown in FIG. 47, asurface topography 4701 is input, and an estimated surface topography isstored in data storage means 4702.

The improved STL-PTC performs scanning using an output signal (error)during an FWS as a command value for a BWS. The improved STL-PTC thusrequires a signal generator 4703 including a stack memory in which dataobtained at the end of the FWS is saved as the first data for the BWS.The block diagram shown in FIG. 47 is characterized in that acalculation 4704 includes a discretized nominal plant P_(n)[z].Installation of such a disturbance estimation mechanism matches thedynamics of an output signal during the FSW with the dynamics of anoutput signal during the BWS.

FIG. 48 is a diagram showing the details of the signal generator 4703.

As shown in FIG. 48, a switch 1 (SW1) for the FWS is kept on for T (=thenumber N_(d) of stages in the memory×the sampling period T_(y)) seconds.An output end conversion value:P(s){circumflex over (d)}  [Expression 26]for a disturbance estimated value obtained by the disturbance estimationmechanism shown in FIG. 47 passes through a stack memory 4801 and isstored in a stack memory 1301. The output end conversion value:P(s){circumflex over (d)}  [Expression 27]stored in the stack memory passes through a sensitivity function 4802:

$\begin{matrix}\frac{1}{1 + {CP}} & \left\lbrack {{Expression}\mspace{14mu} 28} \right\rbrack\end{matrix}$and is thus converted into an output signal for the BWS. During the BWS,a switch 2 (SW2) is turned on to allow the signal generator to generatea target trajectory allowing the error to be adjusted to 0. Thus, thePTC reduces the possible error in a feedforward manner.

FIG. 5 shows the steps of control in the AFM according to the presentinvention. Switching is controlled by observing the X scan waveform asshown in FIG. 5. As shown in FIG. 5, the X scan waveform is triangular.Each image is measured during an FWS 1101 and during a BWS 1102. Duringthe FWS, the switch 1 (SW1) in FIG. 48 is kept on for T (=the numberN_(d) of stages in the memory×the sampling period T_(y) of an outputsignal) seconds (switch-on and switch-off are shown by the height of agraph shown at reference numeral 1103). Error signals are stored in thesignal generator 4703 composed of the stack memory 4801 shown in FIG.48. During the BWS, the switch 2 (SW2) is turned on to allow the signalgenerator to generate a target trajectory allowing the error to beadjusted to 0. Then, the PTC adjusts the error to 0. Thus, the N_(d)memory rows can serve as a feedforward compensator. This enablespossible error signals to be reduced for every sample point. However,during the FWS, the signal generator provides no output.

In FIG. 5, reference numerals 1104, 1106, and 1108 denote learningprocesses. Reference numerals 1105, 1107, and 1109 denote controlprocesses.

Here, since the controlled object is of a second order, when the statevariable (x) is:x=[y,{dot over (y)}]  [Expression 29]the signal generator 4703 for error signals can be designed as shown inFIG. 48. Here, a speed command value:{dot over (r)}[i]  [Expression 30]is as shown in formula (12).(Surface Topography Learning Observer (STLO))

The STL-PTC according to the present embodiment adjusts the trackingerror to 0 at intervals of the sampling period T_(r) (=nT_(u)) for thecommand value to enable the tracking capability of the manipulatingquantity u(t) to be equivalently improved. However, the STL-PTC includesa complicated control mechanism and the dynamics of the plant P_(n)[z].Thus, disadvantageously, a learning signal does not perfectly match theerror signal during the BWS.

Thus, the present embodiment also uses a surface topography learningobserver (STLO). The STLO applies the reverse system of a discretizedplant to estimate a disturbance, and reduces the possible disturbance ina feedforward manner without affecting the stability of the system.Thus, a surface image observed with the AFM can be accurately estimated.

Furthermore, during the FWS, the STLO, like the STL-PTC, need notestimate the disturbance in real time. Thus, creating the reverse systemof the plant delayed by one sample enables the disturbance (d) to bereduced at intervals of the control period T_(u) (0.1 msec). However, itshould be noted that the disturbance is estimated by a zeroth-orderdisturbance observer, so that for any waveform, the disturbancetheoretically delayed by one sample cannot always be estimated.

FIG. 49 is a block diagram showing control performed by the STLO.

In the estimation blocks of the STLO shown in FIG. 49, a surfacetopography 4901 is input, and an estimated surface topography is storedin data storage means 4902. The block diagram shown in FIG. 49 ischaracterized in that a calculation 4904 includes the reciprocal P⁻¹_(n)[z] of the discretized nominal plant P_(n)[z].

In the STLO, the disturbance that estimated with one-sample delay basedon the reverse system of the discretized plant;{circumflex over (d)}  [Expression 31]allows the SW1 to be kept on for T seconds during the FWS. The relevantdata is saved to a stack memory 4903. During the BWS, the stack memorydelays the output of the estimated disturbance by one sample. Thisenables the disturbance:{circumflex over (d)}  [Expression 32]during the BWS to be reduced for every sample point. In the STLO, if anerror or a modeling error following compensation or a disturbance notpresent during the FWS is input during the BWS, this is compensated forby the feedback controller C[z].

Here, provided that the modeling error is small, the disturbance:{circumflex over (d)}  [Expression 33]estimated with one-sample delay is as follows:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 34} \right\rbrack & \; \\{\hat{d} = {{\frac{P(s)}{{zP}_{n}\lbrack z\rbrack}\left( {1 + \Delta} \right)d} \approx {\frac{1}{z}d}}} & (18)\end{matrix}$where Δ denotes a modeling error.

The reverse system of the nominal plant discretized with one-sampledelay using the zeroth-order hold is derived from a discrete-time model.Using a (z) conversion for formulas (8), (9), and (10) results in apulse transfer function expressed by:[Expression 35]G[z]=c _(s)(zI−A _(c))⁻¹ b _(s)  (19)According to formula (19), formula (17) is discretized, and the resultis multiplied by 1/z. Then, the following is given.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 36} \right\rbrack & \; \\{\frac{1}{{zP}_{n}(z)} = \frac{z^{2} + {1.163z} + 0.3978}{{8.907z^{2}} + {5.231z}}} & (20)\end{matrix}$Sampling time is 0.1 msec.

In the present embodiment, a low pass filter Q[z] with no phase delay isintroduced in order to cut noise. This is called a Q filter (Non-PatentDocument 16). When the input of:{circumflex over (d)}  [Expression 37]is defined as r[k] and an output from the Q filter is defined r_(f)[k],a relation expression expressed by formula (21) holds true.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 38} \right\rbrack & \; \\{{r_{f}\lbrack k\rbrack} = {\frac{z + \gamma + z^{- 1}}{\gamma + 2}{r\lbrack k\rbrack}}} & (21)\end{matrix}$

Furthermore, a frequency response from the Q filter is shown in FIGS. 50and 51.

(Simulation)

FIGS. 52 and 53 show the results of simulation in which a rectangularwave-like sample surface is scanned. In FIGS. 52 and 53, the periodbefore 0.02 sec corresponds to the results of simulation of theconventional method, whereas the period after 0.02 sec corresponds tothe results of simulation of the improved STL-PTC.

Here, FIG. 52 shows a temporal variation in the manipulating quantityu(t) of the piezo. FIG. 53 shows a temporal variation in output signaly(t).

Furthermore, FIGS. 54 and 55 show the results of simulation in which arectangular wave-like sample surface is scanned. In FIGS. 54 and 55, theperiod before 0.02 sec corresponds to the results of simulation of theconventional method, whereas the period after 0.02 sec corresponds tothe results of simulation of the STLO.

Here, FIG. 54 shows a temporal variation in the manipulating quantityu(t) of the piezo. FIG. 55 shows a temporal variation in output signaly(t).

In the improved STL-PTC, the sampling time T_(r) for the command value(the signal saved to the stack memory for learning) is 0.2 msec(milliseconds (10⁻³ seconds)). Thus, the compensation fails to beachieved between the sample points for the sampling time T_(y) (0.1msec) for the output signal. Consequently, in FIG. 53, an error occursafter 0.02 sec. Moreover, the disturbance forms a step to affect theplant. Thus, the signal:P(s){circumflex over (d)}  [Expression 39]learned during the FWS disadvantageously degrades the signal for theBWS. However, six Q filters are installed along the target trajectory toreduce this adverse effect.

In contrast, the STLO performs control such that the disturbance:{circumflex over (d)}  [Expression 40]is cancelled at the same time T_(y) as that for the output signal,preventing a possible error between the sample points. Thus, suchresults as shown in FIG. 55 are obtained.(Observation of the Grating Element)

In sample observations with the AFM according to the present embodiment,the sample is, by way of example, a planar brazed holographic gratingstandard article manufactured by Shimadzu Corporation. The gratingelement is shaped like a rectangular wave, and includes a glasssubstrate of resin with grating grooves formed therein. The grooves arecoated with a reflection film of Al or the like. FIG. 56 shows the shapeand size of a grating element 5601 observed with the AFM according tothe present embodiment.

FIG. 64 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the conventional method.

FIG. 65 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the STO.

FIG. 66 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the improved STL-PTC.

FIG. 67 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the STLO.

FIG. 68 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the conventional method.

FIG. 69 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the STO.

FIG. 70 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the improved STL-PTC.

FIG. 71 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the STLO.

In FIGS. 65 and 69, the poles of the low pass filter of the STO are at2,000 Hz. Furthermore, in FIG. 64 to FIG. 67, the scan range of the AFMis 5.5 μm×5.5 μm (FIG. 64 to FIG. 67 are enlarged views of a scan areaof 3 μm×3 μm). In FIG. 64 to FIG. 71, the scanning speed of the AFM is32.2 μm/sec.

FIG. 72 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the conventional method.

FIG. 73 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the STO.

FIG. 74 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the improved STL-PTC.

FIG. 75 shows an image obtained by allowing the AFM to scan the surfaceof the grating element 5601 using the STLO.

FIG. 76 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the conventional method.

FIG. 77 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the STO.

FIG. 78 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the improved STL-PTC.

FIG. 79 shows a sectional waveform obtained by allowing the AFM to scanthe surface of the grating element 5601 using the STLO.

In FIGS. 73 and 77, the poles of the low pass filter of the STO are at2,000 Hz. Furthermore, in FIG. 72 to FIG. 75, the scan range of the AFMis 5.5 μm×5.5 μm (FIG. 72 to FIG. 75 are enlarged views of a scan areaof 3 μm×3 μm). In FIG. 72 to FIG. 79, the scanning speed of the AFM is322 μm/sec.

A comparison of FIG. 64 (FIG. 68) with FIG. 72 (FIG. 76) indicates thatwith the conventional method, increasing the scanning speed of the AFMsignificantly degrades the image. Furthermore, a comparison of FIG. 72(FIG. 76) with FIG. 73 (FIG. 77) indicates that with the STO, thedegradation of the image is reduced even with an increase in thescanning speed of the AFM. However, FIG. 73 indicates that when thescanning speed of the AFM is increased for high-speed scanning, theneven with the STO, the high-speed scanning causes the trackingcapability of the u(t) to be significantly degraded. In this case, themodeling error increases, preventing the rectangular wave-like shapefrom being accurately shaped.

In contrast, FIG. 74 (FIG. 78) indicates that the surface topographyimaged by the improved STL-PTC is nearer rectangular than that imaged bythe STO. This is expected to be because the feedforward compensationreduces possible error signals to improve the tracking capability of theu (t). However, even with the improved STL-PTC, the modeling error andthe inter-sample-point response make the surface topography in the imageappear larger than the actual one having the desired pitch.

On the other hand, the STLO cancels the disturbance in a feedforwardmanner based on the estimated one. Thus, the STLO compensates for themodeling error through feedback to improve the tracking capability ofthe u(t) compared to the improved STL-PTC. Consequently, FIG. 75 (FIG.79) shows that the STLO allows the rectangular wave-like shape to beaccurately imaged.

To quantitatively evaluate the reduction of possible error signals, FIG.57 shows ±3α for the conventional method, the improved STL-PTC, and theSTLO. The standard deviation indicates evaluation for error signalsobtained through about 100 scans.

FIG. 58 to FIG. 63 show waveforms each obtained by superimposing theabove-described error signals on one another.

FIG. 58 to FIG. 60 show error signals obtained when the scanning speedof the AFM is 32.2 μm/sec. FIG. 58 shows error signals obtained when theAFM uses the conventional method. FIG. 59 shows error signals obtainedwhen the AFM uses the improved STL-PTC. FIG. 60 shows error signalsobtained when the AFM uses the STLO.

FIG. 61 to FIG. 63 show error signals obtained when the scanning speedof the AFM is 322 μm/sec. FIG. 61 shows error signals obtained when theAFM uses the conventional method. FIG. 62 shows error signals obtainedwhen the AFM uses the improved STL-PTC. FIG. 63 shows error signalsobtained when the AFM uses the STLO.

As shown in FIG. 57, when the scanning speed of the AFM is 32.2 μm/sec,the improved STL-PTC improves the ±3α by 54.6% compared to theconventional method. The STLO improves the ±3α by 68.1% compared to theconventional method. Furthermore, when the scanning speed of the AFM is322 μm/sec, the improved STL-PTC improves the ±3α by 69.8% compared tothe conventional method. The STLO improves the ±3α by 81.0% compared tothe conventional method.

Third Embodiment

As described below, the present embodiment includes a simpleidentification method for the STLO which uses a low-order model to allowthe frequency characteristics of a plant to be easily identified, and anSTLO improved by using zeroth-order phase error inverse model (ZPEI).

(Internal Configuration of the AFM)

The AFM according to the present embodiment is a special model of aJSPM-5200 manufactured by JEOL Ltd. However, this is only an example,and any AFM is applicable provided that the present embodiment can beincorporated into the AFM. Alternatively, dSPACE1104 may be used tomodify the control mechanism for the AFM so that the control mechanismallows the present embodiment to be implemented.

FIG. 3 is a block diagram showing the flow of signals inside the AFMaccording to the present embodiment.

As shown by reference numeral 310 in FIG. 3, when a sample surface 103is scanned, the displacement of the tip 102 of the cantilever isdetected by a PD (four-piece PhotoDiode) 105 and is output as a signal.The signal is subjected to AD conversion by an AD (AD converter) 305.The resulting signal is input to a DSP (Digital Signal Processor) as y[i].

First, an output x [V] from the PD (four-piece PhotoDiode) is providedaccording to a force curve (the relation expression between a forceexerted on the tip of the cantilever and the distance between thecantilever tip and the sample). In the present embodiment, the output x[V] from the PD can be converted into the displacement y [nm] of thecantilever using the conversion expression y [nm]×K_(PD) [V/nm]=x [V].In the present embodiment, based on the force curve, K_(PD)=4.2804×10⁻²[V/nm].

In the present embodiment, K_(PD) is determined byfirst-order-approximating the measurement data of the force curve(Non-Patent Document 2) obtained with the JSPM-5200, by way of example.

Furthermore, as shown by reference numeral 320 in FIG. 3, a manipulatingquantity u[i] resulting from DA conversion by a DA (DA converter) 301 inthe DSP is amplified by an amplifier 302. The amplified manipulatingquantity u[i] is applied to a PZT (piezo) 107 as a driving voltage. Again provided by the amplifier 302 is K_(g)=20, and the rate ofelongation of the PZT subjected to the driving voltage is K_(PZT)=15.59[nm/V]. That is, the calculation u [V]×K_(PZT) [nm/V] allows the u [V]to be converted into the displacement of the piezo.

Additionally, the gain of the AD/DA in the DSP is adjusted to 1.

In the present embodiment, a motion equation as shown in formula (1) isgiven for the tip 102 of the cantilever with a mass (m), using a modelbased on a contact mode and relating to the interaction between thesample surface 103 and the tip 102 of the cantilever as shown in FIG. 2.

The model may use recesses and protrusions on the sample surface 103 asan input disturbance according to the method described in Non-PatentDocument 2. In the present embodiment, the displacement (y) of thecantilever is measured using the photodiode and laser light. Thus, thetransfer function for the plant is multiplied by a given gain (g). Inthe present embodiment, formula (1) is identified as described below.

(Simple Identification Method)

The simple identification method according to the present embodimentperforms fitting based on a frequency response from a standardsecond-order system determined on the basis of frequency characteristicsidentified when an identification input is a swept sine. Theidentification algorithm of the simple identification method will bedescribed.

(Identification Algorithm)

A transfer function for the standard second-order system is:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 41} \right\rbrack & \; \\{{G(s)} = \frac{\omega_{n}^{2}}{s^{2} + {2{\xi\omega}_{n}s} + \omega_{n}^{2}}} & (22)\end{matrix}$The amplitude value of Equation (22) may be determined. At a peakangular frequency:ω_(p)=√{square root over (1−2ξ²)}ω_(n)  [Expression 42]at which the gain exhibits the maximum value, the peak width M_(p) ofthe gain shown in formula (23) can be obtained.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 43} \right\rbrack & \; \\{M_{p} = \frac{1}{2\xi\sqrt{1 - \xi^{2}}}} & (23)\end{matrix}$

Furthermore, M_(p) is measured based on M_(p)=g_(m)−g_(s) [dB] (formula(24)). Here, g_(m) denotes a peak gain, and g_(s) denotes a DC gain.

The use of M_(p) resulting from formula (24) allows a dampingcoefficient to be uniquely determined based on formula (23).

A general form of the standard second-order system is obtained by makinga program such that the DC gain, the peak gain, and the peak frequencyare automatically acquired from the experimentally obtained frequencyresponse as described above.

According to the above-described procedure, a transfer function for asecond-order model as shown in formula (25) is obtained.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 44} \right\rbrack & \; \\{{P(s)} = \frac{7.209 \times 10^{9}}{s^{2} + {1086s} + {1\text{,}192 \times 10^{9}}}} & (25)\end{matrix}$

For comparison, a high-order (fourth-order) model is used which isidentified based on frequency characteristics acquired by a servoanalyzer, using an invfreqs command provided in Matlab (registered trademark) (Signal Processing Toolbox).

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 45} \right\rbrack & \; \\{{P(s)} = \frac{{b_{0}s^{2}} + {b_{1}s} + b_{2}}{s^{4} + {a_{0}s^{3}} + {a_{1}s^{2}} + {a_{2}s} + a_{3}}} & (26)\end{matrix}$Here, b₀=9.46×10⁸, b₁=1.149×10¹², b₂=2.809×10¹⁸, a₀=8414, a₁=1.648×10⁹,a₂=6.378×10¹², a₃=4.387×10¹⁷.

FIGS. 80 and 81 show the frequency characteristics (dotted line) of thelow-order (second-order) model obtained by the simple identificationmethod and the frequency characteristics (solid line) obtained by theservo analyzer. FIGS. 82 and 83 show the frequency characteristics(dotted line) of the high-order (fourth-order) model obtained by theinvfreqs command and the frequency characteristics (solid line) obtainedby the servo analyzer. A comparison of FIG. 80 (81) with FIG. 82 (83)indicates that the low-order model obtained by the simple identificationmethod can accurately approximate the high-order model obtained by theinvfreqs command.

(Design of the Controller)

The controller according to the conventional method to be compared withthe embodiment of the present invention is a phase delay compensatorused in a product. A transfer function for the controller according tothe conventional method is as shown in formula (3).

Here, the controller according to the conventional method is tuned suchthat the gain margin and the phase margin are 12.2 [dB] and 83.7 [deg],respectively, and that the proportional gain k_(p)=64, ω_(c)=2πf_(c)(f_(c)=0.5 [Hz]), and the cutoff frequency of an open loop=206 [Hz].

(STLO Using a Zeroth-Order Phase Error Inverse Model (ZPEI) (Non-PatentDocument 23))

The STLO using the ZPEI compensates for the disadvantages of a singledirection-surface topography learning observer (SD-STLO) describedbelow.

The single direction-surface topography learning observer (SD-STLO)provides surface topography data obtained during a forward scan (FWS)along a scan path shown in FIG. 7, as a feedforward signal for abackward scan (BWS). Thus, the accuracy with which the surfacetopography is tracked is improved. A control mechanism for the SD-STLDis configured such that the switches SW shown in FIG. 84 are allswitches SW1. That is, the control mechanism is the same as that shownin FIG. 49. However, on the assumption that the discretized plant hasthe minimum phase, when the inverse system of the discretized plant iscreated, the disturbance during the FWS is estimated to be:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 46} \right\rbrack & \; \\{{\hat{d}\lbrack k\rbrack} = {{{\frac{1}{{zP}_{n}\lbrack z\rbrack}{y\lbrack k\rbrack}} - {\frac{1}{z}{u\lbrack k\rbrack}}} = {\frac{1}{z}{d\lbrack k\rbrack}}}} & (27)\end{matrix}$Thus, a learned disturbance d_(stack) [k] output by the stack memory hasa data structure in which:{circumflex over (d)}[k]  [Expression 47]is reversed in the direction of a time axis at the beginning of the BWS.Consequently, d_(stack)[k]=zd[k]. For the BWS, a learned disturbanceu_(ff)[k]=z⁻¹d_(stack)[k] corrected for a phase lead perfectly matchesthe actual disturbance. However, d[k] is approximate to the actualdisturbance subjected to zeroth-order hold.

The SD-STLO can be adapted for non-periodic disturbances (Non-PatentDocument 22) but is not applicable to a discrete-time non-minimum phaseplant owing to the use of the inverse system in discrete time. Whendiscretized using a zeroth-order hold, a continuous-time model with atleast a third relative order may result in unstable zeroes (Non-PatentDocument 24). Thus, the SD-STLO cannot be designed for high-ordermodels.

For example, when a model identified by invfreqs is discretized atT_(s)=0.05 [ms], formula (28) is given.

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 48} \right\rbrack & \; \\{{P\lbrack z\rbrack} = \frac{{b_{0}^{\prime}z^{3}} + {b_{1}^{\prime}z^{2}} + {b_{2}^{\prime}z} + b_{3}^{\prime}}{z^{4} + {a_{0}^{\prime}z^{3}} + {a_{1}^{\prime}z^{2}} + {a_{2}^{\prime}z} + a_{3}^{\prime}}} & (28)\end{matrix}$b′ ₀=1.319, b′ ₁=3.95, b′ ₂=3.75, b′ ₃=1.067a′ ₀=−0.7179, a′ ₁=1.188, a′ ₂=−0.5511, a′ ₃=0.6566  [Expression 49]The model thus has unstable zeros z₀=−1.2375±0.1560i.

Thus, in the present embodiment, to allow a stable inverse model to bedesigned even for a high-order model, a zeroth-order phase error inversemodel (ZPEI) is used to allow the STLO to be applied to the high-ordermodel.

The zeroth-order phase error inverse model (ZPEI) is derived as follows.

Now, the controlled object discretized using the zeroth-order hold isexpressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 50} \right\rbrack & \; \\{{P\left\lbrack z^{- 1} \right\rbrack} = \frac{z^{- d}{B\left\lbrack z^{- 1} \right\rbrack}}{A\left\lbrack z^{- 1} \right\rbrack}} & (29)\end{matrix}$where z⁻¹ denotes a delay operator. In polynomials of a denominator anda numerator in formula (29),B[z ⁻¹ ]=b′ ₀ +b′ ₁ z ⁻¹ + . . . +b′ _(m) z ^(−m) , b′≠0  [Expression51]A[z ⁻¹]=1+a′ ₀ z ⁻¹ + . . . +a′ _(n) z ^(−n)  [Expression 52]d=n−m. Here, A[z⁻¹] is assumed to be stable. To allow unstable zeroes tobe dealt with, the numerator of the controlled object is divided intotwo portions including stable zeros and unstable zeros, respectively.[Expression 53]B[z ⁻¹ ]=B ⁻ [z ⁻¹ ]B ⁺ [z ⁻¹]  (30)

Here, B⁻[z⁻¹] is an sth-order monic polynomial having unstable zeros andstable limit zeros as roots. B⁺[z⁻¹] is a (m−s)th-order polynomialhaving stable zeros as roots. In this case, the zeroth-order phase errorinverse model (ZPEI) of the plant is expressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 54} \right\rbrack & \; \\{{P_{ZPIE}\left\lbrack z^{- 1} \right\rbrack} = {\frac{{A\left\lbrack z^{- 1} \right\rbrack}{B^{-}\lbrack z\rbrack}}{{B^{+}\left\lbrack z^{- 1} \right\rbrack}\left( {B^{-}\lbrack 1\rbrack} \right)^{2}}z^{- s}}} & (31)\end{matrix}$Here, B⁻[z]=B−[z⁻¹]z^(s).

If formula (31) is applied to a tracking control system, the expressionis placed before the controlled object, and a future value r[k+s+d] of atarget value advanced by [s+d] steps is given. Then, zero phasecharacteristics are obtained in which a transfer function from a setpoint (r) to an output (Y) avoids undergoing phase delay at all thefrequencies (Non-Patent Document 23). If the SD-STLO is applied to theZPEI, no future value needs to be given. This is because the learningsignal during the FWS delayed by [s+d] steps can be corrected during theBWS. In this case, the control mechanism for the STLO is such that theswitches shown in FIG. 84 are all switches SW2. The control algorithm ofthe SD-STLO using the ZPEI will be described below.

First, stable zeros and unstable zeros in formula (28) are determined.An output equation from a disturbance d[k] to an estimated disturbance:{circumflex over (d)}[k]  [Expression 55]is expressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 56} \right\rbrack & \; \\{{\hat{d}\lbrack k\rbrack} = {\frac{{B^{-}\left\lbrack z^{- 1} \right\rbrack}{B^{-}\lbrack z\rbrack}}{\left( {B^{-}\lbrack 1\rbrack} \right)^{2}}z^{- {({s + d})}}{d\lbrack k\rbrack}}} & (32)\end{matrix}$For a disturbance:{circumflex over (d)}[k]  [Expression 57]estimated with a delay of s+d steps, during the FWS, S_(FW) in FIG. 84is kept on for T_(FW) seconds. The relevant data is stored in the stackmemory. Then, during the BWS, the stack memory outputs a signalexpressed as:

$\begin{matrix}\left\lbrack {{Expression}\mspace{14mu} 58} \right\rbrack & \; \\{{d\lbrack k\rbrack} = {\frac{{B^{-}\left\lbrack z^{- 1} \right\rbrack}{B^{-}\lbrack z\rbrack}}{\left( {B^{-}\lbrack 1\rbrack} \right)^{2}}z^{({s + d})}{d\lbrack k\rbrack}}} & (33)\end{matrix}$Thus, the estimated disturbance matches the actual disturbance d[k] onlyin a low frequency region where u_(ff)[k]=z^(−(s+d))d_(stack) [k] isbased on formula (33). Thus, the actual disturbance during the BWS isreduced at sample points based on control periods only in the lowfrequency region. For a high frequency region, as shown in FIGS. 87 and88, a reduced gain prevents the learned disturbance from matching theactual disturbance. Switching timings are controlled based on theobservation of X scan waveforms as shown in FIG. 5.(Comparison of the SD-STLO with the Second-Order Model and Fourth-OrderModel)

The ratio of the actual disturbance (d) and the learned disturbanceu_(ff) and the frequency characteristics of u_(ff)/d in the second-ordermodel are compared with those in the fourth-order model.

(Comparison Through Simulation)

FIGS. 85 to 88 show the results of simulation of u_(ff)/d.

FIGS. 85 and 86 show the results of simulation of a frequency responsefrom the u_(ff)/d during the BWS observed when the second-order modelidentified by the simple identification method according to the presentembodiment is used in the SD-STLO. FIGS. 85 and 86 show that when thesecond-order model identified by the simple identification methodaccording to the present embodiment is used in the SD-STLO, the nominalplant can be estimated without a decrease in gain or a phase delay.

FIGS. 87 and 88 show the results of simulation of a frequency responsefrom the u_(ff)/d during the BWS observed when the fourth-order modeland the ZPEI are used in the SD-STLO. FIGS. 87 and 88 show no phasedelay but a decrease in gain in the high frequency region, indicatingthe characteristics of the zeroth-order phase error inverse model(ZPEI).

(Comparison Through Experiments)

FIG. 89 to FIG. 92 show the u_(ff)/d in the actual AFM.

FIGS. 89 and 90 show the results of a frequency response from theu_(ff)/d during the BWS observed when the second-order model identifiedby the simple identification method according to the present embodimentis used in the SD-STLO. Gain characteristics shown in FIGS. 89 and 90show an increase in gain from about 4 [kHz] to 7 [kHz]. This is expectedto reflect the impact of the modeling error between the second-ordermodel P_(n)(s) shown in FIGS. 80 and 81 and the actual controlledobject. Phase characteristics also reflect the impact of the modelingerror and exhibit a significant delay in high frequency.

FIGS. 91 and 92 show the results of a frequency response from theu_(ff)/d during the BWS obtained from the actual AFM when thefourth-order model and the ZPEI are used in the SD-STLO.

FIGS. 91 and 92 indicate that the fourth-order model reduces the impactof the modeling error and that the results are similar to those of thesimulation.

As described above, the discrete-time minimum phase plant for thelow-order model is prevented from suffering a decrease in gain in thehigh frequency region. However, in this case, the modeling error mayvary the estimated disturbance:{circumflex over (d)}  [Expression 59]thus significantly varying the frequency characteristics of theu_(ff)/d. The high-order model is not substantially affected by themodeling error but suffers a decrease in gain in the high frequencyregion. As a result, the estimated disturbance:{circumflex over (d)}  [Expression 60]may be degraded.(Measurement of the Sample)

The results of measurement will be described below in which the AFMaccording to the present embodiment is used to measure a sample that wasa planar brazed holographic grating standard article (grating element)manufactured by Shimadzu Corporation. A grating element 1801 shown inFIGS. 18A, 18B, which is measured with the AFM according to the presentembodiment is shaped like saw teeth and includes a glass substrate ofresin with grating grooves formed therein. The grooves are coated with areflection film of aluminum or the like.

FIG. 93 shows a three-dimensional image of the sample surface obtainedwhen the surface of the grating element 1801 is scanned by using theconventional method for the AFM according to the present embodiment.

FIG. 94 shows a three-dimensional image of the sample surface obtainedwhen the surface of the grating element 1801 is scanned using thesecond-order model in the SD-STLO for the AFM.

FIG. 95 shows a three-dimensional image of the sample surface obtainedwhen the surface of the grating element 1801 is scanned using thefourth-order model and ZPEI in the SD-STLO for the AFM.

In the measurements shown in FIG. 93 to FIG. 95, the scan range is 3μm×3 μm. FIG. 93 to FIG. 95 are centrally enlarged views of the sampleover the range of 5.5 μm×5.5 μm. Furthermore, in the measurements shownin FIG. 93 to FIG. 95, the scanning speed is 322 μm/s, and time requiredfor the whole scan is about 20 seconds.

As shown in FIG. 93 to FIG. 95, when used for the AFM, the SD-STLO usingthe second-order model or the fourth-order model enables a reduction inthe degradation of three-dimensional images of the sample surfacecompared to the conventional method.

FIG. 96 shows superimposed error signals (errors) obtained when thesurface of the grating element 1801 is scanned by using the conventionalmethod for the AFM according to the present embodiment.

FIG. 97 shows superimposed error signals (errors) obtained when thesurface of the grating element 1801 is scanned by using the second-ordermodel in the SD-STLO for the AFM.

FIG. 98 shows superimposed error signals (errors) obtained when thesurface of the grating element 1801 is scanned by using the fourth-ordermodel and ZPEI in the SD-STLO for the AFM.

As shown in FIG. 96 to FIG. 98, when used for the AFM, the SD-STLO usingthe second-order model or the fourth-order model enables a reduction inerror signal (error) compared to the conventional method.

That is, adopting the SD-STLO using the second-order model or thefourth-order model in the AFM can reduce error signal (the impact of thedisturbance), and improve the tracking capability of control input withrespect to the surface topography.

Furthermore, the error signals can be quantitatively evaluated asfollows. When the scanning speed is 32.2 μm/s, the ±3σ for theconventional method is 62.4 [nm], the ±3σ for the STLO is 38.6 [nm], andthe ±3σ for the STLO using the ZPEI is 40.3 [nm]. Thus, both the ±3σ forthe STLO and the ±3σ for the STLO using the ZPEI are smaller than thatfor the conventional method. Consequently, the STLO and the STLO usingthe ZPEI allow the shape topography to be measured more accurately thanthe conventional method.

As described above, the STLO using the ZPEI allows designing of aninverse model that is stable even for a discrete-time non-minimum phaseplant. This enables even a high-order model to provide a controlalgorithm similar to that provided by a low-order model. Thus, when usedfor the AFM, the STLO using the ZPEI allow the topography of the sampleto be measured more accurately than the conventional method.

Fourth Embodiment

The present embodiment includes PLS-STLPTC as described below.

(Pre-Line Scanning Surface Topography Learning with PTC (PLS-STLPTC))

The pre-line scanning surface topography learning with PTC (PLS-STLPTC)according to the present embodiment compensates for the disadvantages ofsingle-direction scanning surface topography learning with PTC(SD-STLPTC) (Non-Patent Document 11) described below.

For the SD-STLPTC, a disturbance estimation mechanism (FIG. 47) needs tobe installed so as to minimize the difference between the actual signaland the learning signal. As shown in FIG. 48, a switch 1 (SW1) for anFWS is kept on for T (=the number N_(d) of stages in a memory×thesampling period T_(y)) seconds. An output end disturbance:P(s){circumflex over (d)}  [Expression 61]estimated by the disturbance estimation mechanism is stored in a stackmemory 4801. The stored disturbance:P(s){circumflex over (d)}  [Expression 62]passes through a sensitivity function 4802:

$\begin{matrix}\frac{1}{1 + {CP}} & \left\lbrack {{Expression}\mspace{14mu} 63} \right\rbrack\end{matrix}$and is thus converted into an output signal for the BWS. During the BWS,a switch 2 (SW2) is turned on to allow a signal generator to generate atarget trajectory allowing the error to be adjusted to 0. Thus, the PTCreduces the possible error in a feedforward manner.

Here, since the controlled object is of a second order, when the statevariable (x) is:x=[y,{dot over (y)}]  [Expression 64]a signal generator for error signals can be designed as shown in FIG.48. However, the speed command value:{dot over (r)}[i]  [Expression 65]is as shown in formula (12).

FIG. 99 to FIG. 102 show the results of simulation of the actual outputsignal (actual signal) and the leaning signal and of simulation of thedisturbance and the control input; the simulation uses control based onthe above-described SD-STLPTC. FIGS. 99 and 100 show the results ofsimulation of measurement of a rectangular-wave sample. FIGS. 101 and102 show the results of simulation of measurement of a triangular-wavesample. The results of simulation of measurement of the rectangular-wavesample and the results of simulation of measurement of thetriangular-wave sample both indicate that the learning signal does notexactly correspond to the folded-back form of the actual signal. This isbecause the shape of the disturbance varies between the FWS and the BWS,resulting in a difference in dynamics between the actual signal and thelearning signal. That is, to prevent a possible difference in dynamics,the output signal for the BWS needs to be generated from the disturbanceduring the BWS. The method applied to the SD-STLO allows the disturbanceto be estimated. However, conditions for the plant and the degradationof the learning signal in a high frequency region are disadvantageous.Thus, the input end disturbance needs to be converted into an output enddisturbance. At this time, the output signal during the FWS is affectedby the dynamics of the plant P_(n)[z]. Consequently, the learning signalcannot be perfectly matched with the output signal for the BWS.

Thus, in the PLS-STLPTC, the dynamics of the output signal during theFWS perfectly matches the dynamics of the output signal during the BWS.As a result, the above-described problems can be solved.

The PLS-STLPTC sequentially applies information obtained from thepreceding scan line (preceding line) and including information for theFWS and information for the BWS, to the succeeding scan line (succeedingline). That is, for the PLS-STLPTC, provided that the surface topographyin the preceding line is the same as that in the succeeding line, thedynamics of the output signal for the preceding scan line perfectlymatches the dynamics of the output signal for the succeeding scan line.Thus, the PLS-STLPTC allows the output signal for the succeeding scanline to be completely reduced for every sample point (T_(r)). Thisenables the capability of the control input to be improved.

A control mechanism for the PLS-STLPTC is similar to that shown in FIG.12. However, the control algorithm in the present embodiment is not ofthe single direction type but of the pre-line scanning type.Furthermore, FIG. 103 shows the details of the signal generator.

The control algorithm performed by the signal generator in FIG. 103 willbe described below. In FIG. 103, T_(FW) denotes the scanning time duringthe FWS, and the scanning time during the BWS is defined as T_(BW).Furthermore, the memory in this case is of an FIFO type. A switch 1(SW1) is kept on for T_(FW)+T_(BW) (=the number N_(d) of stages in thememory×the sampling period T_(y)) seconds, and during this interval, theoutput signal is learned. Then, on the succeeding line, the learnedoutput signal is output during the same interval as a set point for thePTC, enabling perfect tracking. However, also for the PLS-STLPTC, thespeed command value:{dot over (r)}[i]  [Expression 66]is as shown in formula (12).

FIG. 104 to FIG. 107 show the results of simulation of the disturbanceand the control input and output signals using control based on theabove-described PLS-STLPTC. FIGS. 104 and 105 show the results ofsimulation of measurement of a rectangular-wave sample. FIGS. 106 and107 show the results of simulation of measurement of a triangular-wavesample.

In the simulation in FIG. 104 to FIG. 107, the period before 0.02 [sec]is a learning period when the PLS-STLPTC is not performed. That is,during the period before 0.02 [sec], simulation is performed accordingto the conventional method.

In the simulation shown in FIG. 104 to FIG. 107, feedforward controlbased on the PLS-STLPTC is started at 0.02 [sec]. As shown in FIGS. 105and 107, the feedforward control based on the PLS-STLPTC allows theoutput signal to be reduced for every sample point. Also during thesubsequent scans, learning and control are simultaneously performed toperfectly reduce the disturbance.

Furthermore, FIG. 108 is an enlarged view of FIG. 105. FIG. 109 is anenlarged view of FIG. 107. The results shown in FIGS. 108 and 109indicate that the PLS-STLPTC enables perfect tracking provided that thepreceding scan line has the same shape as that of the succeeding scanline regardless of the shape.

1. An atomic force microscope apparatus imaging a surface topography of a sample in a contact mode, the apparatus comprising: a cantilever having a probe interacting with the sample surface via an atomic force and being subjected to a deflection by the atomic force; laser light provision means for allowing first laser light to enter the cantilever; light detection means for detecting second laser light corresponding to the first laser light reflected by the cantilever; a piezo element on which the sample is placed; a controller inputting an input voltage to the piezo element to control the distance between the sample surface and the probe, detecting the deflection of the cantilever as an output voltage based on a relative change in light intensity in a vertical direction to which a photodiode is subjected owing to the second laser light, then during a forward scan, measuring and storing the surface topography, then during a backward scan of the same line as that for the forward scan, using the stored surface topography to generate a tracking error for the backward scan, and measuring the surface topography of the sample surface based on the tracking error; and data storage means for recording the measured tracking error, wherein the controller uses a means for perfect tracking control in which a sampling period and a control period for a target trajectory generated from the tracking error are different from each other and which uses multirate control, to reduce the tracking error to measure the surface topography of the sample surface.
 2. An atomic force microscope apparatus imaging a surface topography of a sample in a contact mode, the apparatus comprising: a cantilever having a probe interacting with the sample surface via an atomic force and being subjected to a deflection by the atomic force; laser light provision means for allowing first laser light to enter the cantilever; light detection means for detecting second laser light corresponding to the first laser light reflected by the cantilever; a piezo element on which the sample is placed; a controller inputting an input voltage to the piezo element to control the distance between the sample surface and the probe, detecting the deflection of the cantilever as an output voltage based on a relative change in light intensity in a vertical direction to which a photodiode is subjected owing to the second laser light, then during a forward scan, measuring and storing the surface topography, and during a backward scan of the same line as that for the forward scan, using the stored surface topography for control to estimate the surface topography of the sample surface; and data storage means for recording the surface topography, wherein the controller uses means for realizing an inverse system of a discretized plant derived from a state equation that discretizes a controlled object of the controller at a sampling period for a target trajectory generated from the tracking error, to estimate the surface topography of the sample surface.
 3. The atomic force microscope apparatus according to claim 2, wherein to estimate the surface topography of the sample surface, the controller uses means for using a zeroth-order phase error inverse model including a polynomial having an unstable zero and a stable limit zero as roots and a polynomial having a stable zero as a root.
 4. The atomic force microscope apparatus according to claim 1, further comprising a servo analyzer acquiring frequency characteristics of a transfer function from the input voltage to the piezo element to the output voltage, wherein the transfer function is automatically identified by a standard second-order system based on a peak gain, a DC gain, and a peak frequency included in the frequency characteristics acquired by the servo analyzer. 